Returning to the CSP motivation
of the subpower membership problem,
the case in which
representing constraint relations by generators
is the most advantageous, is when the finite
algebra \(\mathbf {A}\)  associated to the CSP has the property that
all subalgebras of finite powers \(\mathbf {A}^n\)  of \(\mathbf {A}\) 
admit small generating sets;
here `small' means that the size
is bounded by \(p(n)\)  for some
polynomial \(p\)  depending only on \(\mathbf {A}\) .
These algebras have
been intensively studied within the theory of constraint satisfaction problems,
and
constitute one of the two known major classes of CSPs
for which general polynomial time algorithms were found several years ago
[1]}, [2]}.
It was proved in [3]} that a finite algebra \(\mathbf {A}\)  has the
aforementioned property on small generating sets for subalgebras of powers
if and only if \(\mathbf {A}\)  has few subpowers
in the following sense:
there is a polynomial \(q\)  depending only on \(\mathbf {A}\)  such that
\(\mathbf {A}^n\)  has at most \(2^{q(n)}\)  subalgebras.
This number is indeed `small',
as there exist finite
algebras \(\mathbf {A}\)  (e.g., semilattices or unary algebras) for which
the number of subalgebras of \(\mathbf {A}^n\)  is doubly exponential in \(n\) .
